Place racks in the upper and lower thirds of the oven and preheat the oven to 375 degrees F. Line two baking sheets with parchment paper.
In a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment, beat together the butter and sugar on medium speed until smooth, creamy, and combined, about 1 minute. Add the egg and beat again until it disappears. Beat in the rum extract, vanilla extract, baking powder, nutmeg, and salt.
Sprinkle the whole wheat flour and all-purpose flour over the top. Beat on low speed, mixing just until the flour completely disappears.
Make the coating: In a large ziptop bag, place the sugar and cinnamon. Seal and shake to combine.
Scoop the dough into 1-inch-diameter balls and drop them into the bag about 6 at a time. Gently roll/toss the balls in the cinnamon sugar until each is completely coated. Transfer to the prepared baking sheets, spacing at least 1 1/2 inches apart. With the flat bottom of a drinking glass, gently flatten the cookies until they are 1/2 inch thick and 1 1/2 inches wide.
Bake the cookies on the upper and lower racks of the oven for 8 to 9 minutes, switching the pans’ position once halfway through if your oven has hot spots. Remove the pans from the oven and place them on a wire rack. Let cool on the pan for 3 minutes, then gently transfer them to the wire rack to finish cooling completely.
For the chocolate dip: Line a fresh baking sheet with clean parchment or wax paper. Place the white and semisweet chocolate in separate microwave-safe bowls. Add 1 1/2 teaspoons coconut oil to each. Microwave for 30 seconds, stir, then return to the microwave and continue heating in 10- to 15-second bursts, until the chocolate is almost fully melted. Remove from the microwave and stir, allowing the residual heat to melt the chocolate completely, stirring until it is shiny and smooth.
Dip each cookie halfway into the chocolate, then transfer it to the prepared baking sheet. While the chocolate is still wet, add any desired sprinkles. Repeat with the remaining cookies, alternating between white and semisweet chocolate. Let rest until the chocolate sets. Enjoy!
Store the cookies in an airtight container at room temperature for up to 3 days or freeze for up to 1 month.